Download Statistical Methods Chapter 1: Overview and Descriptive Statistics

Statistical Methods
Chapter 1: Overview and Descriptive Statistics
• General Introduction
– Statistics studies data, population, and samples.
– Descriptive Statistics vs Inferential Statistics.
• Descriptive Statistics
– Pictorial and tabular methods
∗ Stemplot, dotplot, histogram, boxplot.
– Numerical measures
∗ Measures of Location: Mean and Median.
∗ Measures of Variability: Range, Variance, and IQR.
• Inferential Statistics
– Draw conclusions about a certain population parameter.
∗ Confidence Intervals.
∗ Hypothesis Testing.
What does statistics study?
Statistics is a mathematical science pertaining collection, presentation, analysis and interpretation
of data.
• Population: a well-defined collection of objects.
• Sample: a subset of the population.
• Variable: characteristics of the objects.
• Observation: an observed value of a variable.
• Data: a collection of observations.
statistics → study data → understand the population
About Variable
What is variable?
Characteristics of a population of interest whose values vary.
A variable can be
• Categorical
– e.g. x = gender of a person (male, female)
• Numerical
– Discrete variable: e.g. x = # of students in a class
– Continuous variable: e.g. x = height of a student
Types of Data
Data come from making observations either on a single variable or simultaneously on two or more
variables.
• Univariate data: observations on a single variable
• Bivariate data: observations on two variables e.g. (x, y) =(height, weight) of a student
• Multivariate data: observations on more than two variables e.g. (x, y, z) = (height, weight, gender) of a student
How to study data?
What is Statistics?
• Data collection
– Sampling methods, experimental design.
• Data analysis, presentation & interpretation
– Descriptive statistics - summarize and describe features of data
∗ Visual methods: dotplot, pie chart, histogram.
∗ Numerical methods: measures of location ( mean, median) and variation (range, variance)
– Inferential statistics - make inference about the population from samples
∗ Point estimate, confidence intervals, hypothesis testing.
Inferential Statistics and Probability Theory
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Descriptive Statistics: Visual Methods
• Stem-and-leaf display
• Dotplot
• Histogram
• Boxplot
Stem-and-leaf Display
Example 1
The number of touchdown passes thrown by each of the 31 teams in the National Football League in
2000 is given below: {14, 29, 22, 18, 20, 15, 6, 9, 32, 18, 19, 18, 23, 28, 37, 21, 14, 19, 21, 20, 16, 22,
33, 28, 12, 18, 22, 14, 33, 21, 12}
• What does the data tell?
The tens digits called stems are arranged as a column to the left. The ones digits are listed to the right of each
stem and are called leaves.
0|69
1|4858984962842
2|920381102821
3|2733
0|69
1|2244456888899
2|001112223889
3|2337
• What can we say about the data set now?Most teams had 10 − 29 touchdown passes.
Refined Stem-and-leaf Display
When too many leaves are lumped into a few stems, splitting the stem helps reveal more information
about the distribution of data. We can further ”refine” the above stem-and-leaf display by splitting each
stem into two parts: low and high.
0H|69
1L|44242
1H|85898968
2L|203110221
3
2H|988
3L|233
3H|7
• What can we say about the data set now? Most teams had 15 − 24 touchdown passes.
Compare Data by Stem-and-leaf Display
Example 2
Suppose we also have data from the 1998 season. We can compare the numbers of touchdown passes in
the 1998 and 2000.
Year 1998| |Year 2000
7|0H|69
213|1L|44242
987776665|1H|85898968
44331110|2L|203110221
8865|2H|988
332|3L|233
|3H|7
11|4L|
• The peaks of the two seasons are slightly different.
• For both seasons, most teams had 15 − 24 touchdown passes.
• The shapes of the data distributions are similar.
Summary: Stem-and-leaf Display
• How to make a stem-and-leaf display?
1. Select one or more leading digits for the stem values (any value appropriate). The trailing
digits become the leaves.
2. List possible stem values in a vertical column.
3. Put the leaf for each observation besides the corresponding stem.
4. Indicate the units for stems and leaves.
• What can a stem-and-leaf display tell?
– Typical value
– Symmetry of distribution
– Peaks
– Outliers
Stem-and-leaf display is suitable for a data set with a moderate size.
Dotplot
Example 3
O-ring temperatures (F ◦ ) for test firings or actual launches of the shuttle rocket engine. {84, 49, 61, 40,
83, 67, 45, 66, 70, 69, 80, 58, 68, 60, 67, 72, 73, 70, 57, 63, 70, 78, 52, 67, 53, 67, 75, 61, 70, 81, 76,
79, 75, 76, 58, 31}
4
Dotplot of the O-ring temperature data
30
40
50
60
Temperature
70
80
90
Summary: Dotplot
• How to make a dotplot?
1. Represent each obs by a dot above the corresponding location on a measurement scale.
2. Stack dots vertically when a value occurs more than once.
• What can a dotplot tell?
–
–
–
–
Location of typically values
Spread of data set
Extreme values
Gaps between values
Dotplot is a nice display of data when a data set is reasonably small or has only a few distinct values.
Histogram
What if a data set is large?
Use Histogram
For different types of data, we construct histograms differently.
• Histogram for discrete data
• Histogram for continuous data
• Histogram for categorical (qualitative) data, also known as Bar-graph
Histogram for Discrete Data
• Frequency (Count) In a discrete data set, frequency of a value c is the number of occurrences of c
in the data set.
• Relative frequency The relative frequency of a value c is
frequency of c
n
where n is the total number of observations in the data set.
relative frequency of a value c =
If we list frequencies of a data set in a table, it is called frequency distribution/table.
5
Constructing Histogram for Discrete Data
How to create a histogram for a discrete data set?
1. Determine the distinct values c1 , c2 , c3 , . . . , cr in the data set.
2. Calculate the relative frequency for each cj , j = 1, 2, . . . , r:
relative frequency of cj =
number of occurrences of cj
n
3. Mark the cj ’s on a horizontal scale, draw a rectangle whose height is the relative frequency of cj ,
where (j = 1, 2, . . . , r).
The area of the rectangle is proportional to the relative frequency.
Histogram for Discrete Data
Example 4
100 married couples between 30 and 40 years of age are studied to see how many children each couple
have. Table below is the frequency table of this data set.
--------------------------------------------------Kids
# of couples
Relative Freq
--------------------------------------------------0
11
0.11
1
22
0.22
2
24
0.24
3
30
0.30
4
11
0.11
5
1
0.01
6
0
0.00
7
1
0.01
--------------100
1.00
---------------------------------------------------
Histogram of Example 4
Histogram for Continuous Data
How to create a histogram for a continuous data set?
1. Divide the measurement axis into a number of class intervals/classes such that each obs falls into exactly one interval.
Denote these intervals by: I1 , I2 , . . . , Ir .
• To ensure that each obs falls into exactly one interval, we may use intervals in the form: I1 = [a1 , a2 ), I2 =
[a2 , a3 ), . . .
• We may use Ij ’s of the same interval length, this is called equal class width; we may also use Ij ’s of different
interval lengths, this is called unequal class width j = 1, 2, 3, . . . , r.
2. Calculate relative frequency for each interval Ij , j = 1, 2, 3, . . . , r.
3. Draw a rectangle above each Ij .
• For equal class width case, rectangle height = relative frequency.
relative frequency of the class interval Ij
• For unequal class width case: rectangle height =
, the resulting rectclass interval width
angle heights here are called densities.
The area of the rectangle is proportional to the relative frequency. For unequal class width histograms,
the total area of all rectangles is 1.
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Histogram for Continuous Data
Example 5
Adjusted energy consumption during a particular period for a sample of 90 gas-heated homes are
recorded.
We divide the class intervals as follows:
------------------------------------------------------------------------------------------Class
[1, 3) [3, 5) [5, 7) [7, 9) [9, 11) [11, 13) [13, 15) [15, 17) [17, 19)
------------------------------------------------------------------------------------------Freq.
1
1
11
21
25
17
9
4
1
------------------------------------------------------------------------------------------Relative freq. 0.011
0.011 0.122
0.233
0.278
0.189
0.100
0.044
0.011
-------------------------------------------------------------------------------------------
Histogram of Example 5
Histogram Shapes
Histograms have a variety of shapes, the shape of a histogram conveys important information about
the distribution of data.
• Unimodal: Single peak
• Bimodal: Two peaks
• Multimodal: Two more peaks
• Symmetric: Left ≈ right
• Positively skewed: Right tail stretching out
• Negatively skewed: Left tail stretching out
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Histogram Shapes
Descriptive Statistics: Numerical Measures
Visual displays give us general ideas about the shape of data distribution, typical values. Numerical
measures give us quantitative measures instead.
• Measures of location
– Mean
– Median
– Trimmed mean
– Quartiles
• Measures of variability
8
– Variance
– Standard deviation
• Another visual display of data: Boxplot.
Measure of Location: Mean
• Sample mean of a sample of size n {x1 , x2 , . . . , xn } is the arithmetic mean of all obs in the data
set and is denoted by x̄:
Pn
xi
x̄ = i=1
n
– Interpretation of x̄: measures location/center of a sample.
– x̄ takes every individual obs into account and weigh them equally.
• Population mean is ”average/center” point of a population, and is usually denoted by µ.
• Use sample mean x̄ to estimate and make inferences about the usually unknown population mean
µ.
Sample Mean
Example 6
The following sample contains weights (lbs) of basses in a specific lake: {x1 = 1.22, x2 = 1.51, x3 =
1.34, x4 = 1.60, x5 = 0.98, x6 = 1.71, x7 = 1.82, x8 = 1.04, x9 = 1.10, x10 = 0.85, x11 = 1.08}
The mean weight of this sample is:
1.22 + 1.51 + · · · + 1.08
= 1.30
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Suppose we catch another bass in the lake and it weighs 12.52 lbs. {x1 = 1.22, x2 = 1.51, x3 =
1.34, x4 = 1.60, x5 = 0.98, x6 = 1.71, x7 = 1.82, x8 = 1.04, x9 = 1.10, x10 = 0.85, x11 =
1.08, x12 = 12.52} The mean weight of this sample becomes:
x̄ =
x̄ =
1.22 + 1.51 + · · · + 12.52
= 2.23
12
• Drawback: Sample mean is very sensitive to outliers. Alternative measure: Median
Measure of Location: Median
• Sample median of a sample of size n {x1 , x2 , . . . , xn } is the middle value of the sample, denoted
by x̃. It is obtained by:
1. Order the n obs from smallest to largest {x(1) , x(2) , . . . , x(n) }.
2. The median is then:

x̃ =

)
x( n+1
2
when n is odd

 x( n2 ) +x( n2 +1)
when n is even
2
– Interpretation of x̃ : the value in the middle of the sample
– Note that to calculate sample median, only one or two obs in the middle are needed.
• Population median is the middle point in a population, and is usually denoted by µ̃.
• Use sample median x̃ to estimate and make inferences about the usually unknown population
median µ̃.
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Sample Median - Example 6
Before we caught the huge bass, we had n = 11 obs in the sample:
1. Order the data set from smallest to largest: x(1) = 0.85, x(2) = 0.98, x(3) = 1.04, . . . , x(6) =
1.22, . . . , x(11) = 1.82
2. n is odd, so x̃ = x( 11+1 ) = x(6) = 1.22
2
Comparing x̄ = 1.30 and x̃ = 1.22, the difference is not big.
Now after we caught the 12.52-lb fish, our sample size becomes n = 12, and median:
1. Order the data set from smallest to largest: x(1) = 0.85, x(2) = 0.98, x(3) = 1.04, . . . , x(6) =
1.22, x(7) = 1.34, . . . , x(11) = 1.82, x(12) = 12.52
2. n is even, so x̃ =
x(6) +x(7)
2
=
1.22+1.34
2
= 1.28
Median is clearly not severely affected.
Measures of Location: Trimmed Mean
• x̄ is sensitive to outliers, while x̃ is very insensitive to outliers, two extremes.
• A trimmed mean is a compromise between these two.
• Give the number α, where 0 < α < 1, the 100α% trimmed mean is computed by eliminating the
smallest and largest 100α% in the sample and then calculate the average over the obs left in the
sample.
• See details in your textbook (page 28).
Measures of Location: Quartiles
• Median separates the sample into two parts: lower sub-sample and upper sub-sample. n odd:
{x(1) , . . . , x( n+1 ) } and {{x( n+1 ) }, . . . , x(n) } n even: {x(1) , . . . , x( n2 ) } and {{x( n2 +1) }, . . . , x(n) }
2
2
• Quartiles divide the lower and upper sub-samples into two parts:
– 1st Quartile: Q1 = median of the lower sub-sample, also called the lower fourth
– 2nd Quartile: Q2 = median of the entire sample
– 3rd Quartile: Q3 = median of the upper sub-sample, also called the upper fourth
– Inter Quartile Range: IQR = Q3 − Q1 , also called fourth spread
Quartile Example
Still use our bass example, rank the 11 obs: {x(1) = 0.85, x(2) = 0.98, x(3) = 1.04, x(4) =
1.08, x(5) = 1.10, x(6) = 1.22, x(7) = 1.34, x(8) = 1.51, x(9) = 1.60, x(10) = 1.71, x(11) = 1.82}
x(3) + x(4)
= 1.060
2
Q2 = x̃ = 1.22
x(8) + x(9)
Q3 =
= 1.555
2
IQR = 1.555 − 1.060 = 0.495
Q1 =
10
Measures of Variability
Data set 1
{−0.20, −0.10, −0.01, 0, 0.01, 0.10, 0.20}, Sample mean: x̄1 = 0
Data set 2
{−10000, −2000, −100, 0, 100, 2000, 10000}, Sample mean: x̄2 = 0
Two data sets have the same means, but obviously second one is more spread out. So we need numeric
measures of such variability too. Variance is one of such measures.
Measures of Variability: Variance
To compute sample variance for a sample {x1 , x2 , . . . , xn }
1. calculate the sample mean x̄
2. calculate the deviations of each obs from x̄: x1 − x̄, x2 − x̄, . . . , xn − x̄
3. s2 is the average sum of squares of the deviations: s2 =
Pn
2
i=1 (xi −x̄)
n−1
• interpretation: average magnitude of the deviation from the sample mean
√
• Sometimes we also use sample standard deviation: s = s2
• Similarly, we also have population variance σ 2 and population std dev σ as a measure of variability of the population.
• s2 /s could be used to estimate or make inferences about σ 2 /σ.
The Divisor n − 1
Why do we use n − 1 as the divisor to calculate s2 ?
• We hope s2 can be a good estimate of σ 2 , ideally, we want to calculate s2 as:
P
(xi − µ)2
2
s =
n
• µ is something unknown from the population, a replacement of µ is x̄, but obs in a sample tend
to be closer to the sample mean x̄, resulting a relatively smaller sum of squares, so we use n − 1
instead of n as the divisor to compensate for this.
2
• n − 1 is called
P degree of freedom. This is because s is based on n deviations x1 − x̄, . . . , xn − x̄,
but since (xi − x̄) = 0, any n − 1 deviations will be enough.
Properties of s2
A working formula for s2
P 2
Sxx
s2 = n−1
, Sxx =
xi −
P
( xi )2
n
Properties of s2
Let {x1 , x2 , . . . , xn } be the sample and c be any nonzero constant.
• If y1 = x1 + c, y2 = x2 + c, . . . , yn = xn + c, then s2y = s2x .
• If y1 = cx1 , y2 = cx2 , . . . , yn = cxn , then s2y = c2 s2x and sy = |c|sx .
11
Sample Variance
Let us look at the data sets that have the same mean. Data set 1: {−0.20, −0.10, −0.01, 0, 0.01, 0.10, 0.20},
x̄1 = 0
0.202 + 0.102 + 0.012 + 0 + 0.012 + 0.102 + 0.202
s21 =
= 0.017
7−1
Data set 2: {−10000, −2000, −100, 0, 100, 2000, 10000}, x̄2 = 0
s22 =
100002 + 20002 + 1002 + 0 + 1002 + 20002 + 100002
= 3.0 × 107
7−1
Boxplot
Boxplot is very useful in describing several of a data set’s important features such as: center, spread,
symmetry and outliers.
1. Draw a horizontal axis, find Q1 , Q2 and Q3 and calculate IQR.
2. Place a rectangle above the axis, with the left edge at Q1 , right edge at Q3 .
3. Place a vertical line segment inside the rectangle at the location of Q2 .
4. Draw whiskers out from each end of the rectangle to the smallest and largest obs.
Boxplot With Outliers
We can also draw boxplots that show outliers.
• Any obs farther than 1.5IQR from the nearest quartile is a mild outlier
• Any obs farther than 3IQR from the nearest quartile is a extreme outlier
To draw boxplot that show outliers, we modify the boxplot by:
1. Drawing a whisker out from the rectangle to the smallest and largest obs that are not outliers.
2. Plot mild outliers by solid dots, plot extreme outliers with circles. (optional)
Boxplot
Example 7
This is example 1.18 in your textbook (page 37) Pulse width data, n = 25:
{5.30,
88.00,
93.60,
95.80,
99.00,
8.20,
90.20,
94.30,
95.90,
101.40,
13.80,
91.50,
94.80,
96.60,
103.70,
74.10,
92.40,
94.90,
96.70,
106.00,
85.30,
92.90,
95.50,
98.10,
113.50}
We have:
Q1 = 90.2,
Q2 = x̃ = 94.8,
1.5IQR = 9.75,
Q3 = 96.7,
3IQR = 19.5
So, the extreme outliers are:
5.30, 8, 20, 13.80
The mild outliers are:
74.10, 113.5
12
IQR = 6.5
0
20
40
60
Pulse Width
80
Boxplot of Example 7
Distribution Shapes, Boxplots and Measures of Location
13
100
120